The topology on ${\displaystyle X}$  can be described by specifying that an absolutely convex subset ${\displaystyle U}$  is a neighborhood of ${\displaystyle 0}$  if and only if ${\displaystyle U\cap X_{n}}$  is an absolutely convex neighborhood of ${\displaystyle 0}$  in ${\displaystyle X_{n}}$  for every ${\displaystyle n.}$ 

A strict LB-space is complete,^[2] barrelled,^[2] and bornological^[2] (and thus ultrabornological).

If ${\displaystyle D}$  is a locally compact topological space that is countable at infinity (that is, it is equal to a countable union of compact subspaces) then the space ${\displaystyle C_{c}(D)}$  of all continuous, complex-valued functions on ${\displaystyle D}$  with compact support is a strict LB-space.^[3] For any compact subset ${\displaystyle K\subseteq D,}$  let ${\displaystyle C_{c}(K)}$  denote the Banach space of complex-valued functions that are supported by ${\displaystyle K}$  with the uniform norm and order the family of compact subsets of ${\displaystyle D}$  by inclusion.^[3]

Final topology on the direct limit of finite-dimensional Euclidean spaces

Let

${\displaystyle {\begin{alignedat}{4}\mathbb {R} ^{\infty }~&:=~\left\{\left(x_{1},x_{2},\ldots \right)\in \mathbb {R} ^{\mathbb {N} }~:~{\text{ all but finitely many }}x_{i}{\text{ are equal to 0 }}\right\},\end{alignedat}}}$ 

denote the space of finite sequences, where ${\displaystyle \mathbb {R} ^{\mathbb {N} }}$  denotes the space of all real sequences. For every natural number ${\displaystyle n\in \mathbb {N} ,}$  let ${\displaystyle \mathbb {R} ^{n}}$  denote the usual Euclidean space endowed with the Euclidean topology and let ${\displaystyle \operatorname {In} _{\mathbb {R} ^{n}}:\mathbb {R} ^{n}\to \mathbb {R} ^{\infty }}$  denote the canonical inclusion defined by ${\displaystyle \operatorname {In} _{\mathbb {R} ^{n}}\left(x_{1},\ldots ,x_{n}\right):=\left(x_{1},\ldots ,x_{n},0,0,\ldots \right)}$  so that its image is

${\displaystyle \operatorname {Im} \left(\operatorname {In} _{\mathbb {R} ^{n}}\right)=\left\{\left(x_{1},\ldots ,x_{n},0,0,\ldots \right)~:~x_{1},\ldots ,x_{n}\in \mathbb {R} \right\}=\mathbb {R} ^{n}\times \left\{(0,0,\ldots )\right\}}$ 

and consequently,

${\displaystyle \mathbb {R} ^{\infty }=\bigcup _{n\in \mathbb {N} }\operatorname {Im} \left(\operatorname {In} _{\mathbb {R} ^{n}}\right).}$ 

Endow the set ${\displaystyle \mathbb {R} ^{\infty }}$  with the final topology ${\displaystyle \tau ^{\infty }}$  induced by the family ${\displaystyle {\mathcal {F}}:=\left\{\;\operatorname {In} _{\mathbb {R} ^{n}}~:~n\in \mathbb {N} \;\right\}}$  of all canonical inclusions. With this topology, ${\displaystyle \mathbb {R} ^{\infty }}$  becomes a complete Hausdorff locally convex sequential topological vector space that is not a Fréchet–Urysohn space. The topology ${\displaystyle \tau ^{\infty }}$  is strictly finer than the subspace topology induced on ${\displaystyle \mathbb {R} ^{\infty }}$  by ${\displaystyle \mathbb {R} ^{\mathbb {N} },}$  where ${\displaystyle \mathbb {R} ^{\mathbb {N} }}$  is endowed with its usual product topology. Endow the image ${\displaystyle \operatorname {Im} \left(\operatorname {In} _{\mathbb {R} ^{n}}\right)}$  with the final topology induced on it by the bijection ${\displaystyle \operatorname {In} _{\mathbb {R} ^{n}}:\mathbb {R} ^{n}\to \operatorname {Im} \left(\operatorname {In} _{\mathbb {R} ^{n}}\right);}$  that is, it is endowed with the Euclidean topology transferred to it from ${\displaystyle \mathbb {R} ^{n}}$  via ${\displaystyle \operatorname {In} _{\mathbb {R} ^{n}}.}$  This topology on ${\displaystyle \operatorname {Im} \left(\operatorname {In} _{\mathbb {R} ^{n}}\right)}$  is equal to the subspace topology induced on it by ${\displaystyle \left(\mathbb {R} ^{\infty },\tau ^{\infty }\right).}$  A subset ${\displaystyle S\subseteq \mathbb {R} ^{\infty }}$  is open (resp. closed) in ${\displaystyle \left(\mathbb {R} ^{\infty },\tau ^{\infty }\right)}$  if and only if for every ${\displaystyle n\in \mathbb {N} ,}$  the set ${\displaystyle S\cap \operatorname {Im} \left(\operatorname {In} _{\mathbb {R} ^{n}}\right)}$  is an open (resp. closed) subset of ${\displaystyle \operatorname {Im} \left(\operatorname {In} _{\mathbb {R} ^{n}}\right).}$  The topology ${\displaystyle \tau ^{\infty }}$  is coherent with family of subspaces ${\displaystyle \mathbb {S} :=\left\{\;\operatorname {Im} \left(\operatorname {In} _{\mathbb {R} ^{n}}\right)~:~n\in \mathbb {N} \;\right\}.}$  This makes ${\displaystyle \left(\mathbb {R} ^{\infty },\tau ^{\infty }\right)}$  into an LB-space. Consequently, if ${\displaystyle v\in \mathbb {R} ^{\infty }}$  and ${\displaystyle v_{\bullet }}$  is a sequence in ${\displaystyle \mathbb {R} ^{\infty }}$  then ${\displaystyle v_{\bullet }\to v}$  in ${\displaystyle \left(\mathbb {R} ^{\infty },\tau ^{\infty }\right)}$  if and only if there exists some ${\displaystyle n\in \mathbb {N} }$  such that both ${\displaystyle v}$  and ${\displaystyle v_{\bullet }}$  are contained in ${\displaystyle \operatorname {Im} \left(\operatorname {In} _{\mathbb {R} ^{n}}\right)}$  and ${\displaystyle v_{\bullet }\to v}$  in ${\displaystyle \operatorname {Im} \left(\operatorname {In} _{\mathbb {R} ^{n}}\right).}$ 

Often, for every ${\displaystyle n\in \mathbb {N} ,}$  the canonical inclusion ${\displaystyle \operatorname {In} _{\mathbb {R} ^{n}}}$  is used to identify ${\displaystyle \mathbb {R} ^{n}}$  with its image ${\displaystyle \operatorname {Im} \left(\operatorname {In} _{\mathbb {R} ^{n}}\right)}$  in ${\displaystyle \mathbb {R} ^{\infty };}$  explicitly, the elements ${\displaystyle \left(x_{1},\ldots ,x_{n}\right)\in \mathbb {R} ^{n}}$  and ${\displaystyle \left(x_{1},\ldots ,x_{n},0,0,0,\ldots \right)}$  are identified together. Under this identification, ${\displaystyle \left(\left(\mathbb {R} ^{\infty },\tau ^{\infty }\right),\left(\operatorname {In} _{\mathbb {R} ^{n}}\right)_{n\in \mathbb {N} }\right)}$  becomes a direct limit of the direct system ${\displaystyle \left(\left(\mathbb {R} ^{n}\right)_{n\in \mathbb {N} },\left(\operatorname {In} _{\mathbb {R} ^{m}}^{\mathbb {R} ^{n}}\right)_{m\leq n{\text{ in }}\mathbb {N} },\mathbb {N} \right),}$  where for every ${\displaystyle m\leq n,}$  the map ${\displaystyle \operatorname {In} _{\mathbb {R} ^{m}}^{\mathbb {R} ^{n}}:\mathbb {R} ^{m}\to \mathbb {R} ^{n}}$  is the canonical inclusion defined by ${\displaystyle \operatorname {In} _{\mathbb {R} ^{m}}^{\mathbb {R} ^{n}}\left(x_{1},\ldots ,x_{m}\right):=\left(x_{1},\ldots ,x_{m},0,\ldots ,0\right),}$  where there are ${\displaystyle n-m}$  trailing zeros.

Counter-examples edit

There exists a bornological LB-space whose strong bidual is not bornological.^[4] There exists an LB-space that is not quasi-complete.^[4]

• DF-space – class of special local-convex spacePages displaying wikidata descriptions as a fallback
• Direct limit – Special case of colimit in category theory
• Final topology – Finest topology making some functions continuous
• F-space – Topological vector space with a complete translation-invariant metric
• LF-space – Topological vector space

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